Prove that these three statements are equivalent: (a) For every continuous function $f:S^n \to \mathbb R^n$, there exists an $x\in S^n$ such that $f(x)=f(-x)$. (b) There is no antipodal mapping $f:S^n \to S^{n-1}$. (c) For every covering of $S^n$ with closed sets $A_0,\dots,A_n$, there exists an index $i$ such that $A_i\cap -A_i\neq \emptyset$.